2012.72: Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number
2012.72: Awad H. AlMohy, Nicholas J. Higham and Samuel D. Relton (2012) Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number.
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Abstract
The most popular method for computing the matrix logarithm is the inverse scaling and squaring method, which is the basis of the recent algorithm of [A. H. AlMohy and N. J. Higham, \emph{Improved inverse scaling and squaring algorithms for the matrix logarithm}, SIAM J. Sci.\ Comput., 34 (2012), pp.~C152C169]. We show that by differentiating the latter algorithm a backward stable algorithm for computing the Fr\'echet derivative of the matrix logarithm is obtained. This algorithm requires complex arithmetic, but we also develop a version that uses only real arithmetic when $A$ is real; as a special case we obtain a new algorithm for computing the logarithm of a real matrix in real arithmetic. We show experimentally that our two algorithms are more accurate and efficient than existing algorithms for computing the Fr\'echet derivative. We also show how the algorithms can be used to produce reliable estimates of the condition number of the matrix logarithm.
Item Type:  MIMS Preprint 

Additional Information: 

Uncontrolled Keywords:  matrix logarithm, principal logarithm, inverse scaling and squaring method, Fr\'{e}chet derivative, condition number, Pad\'{e} approximation, backward error analysis, matrix exponential, matrix square root, MATLAB, logm. 
Subjects:  MSC 2000 > 15 Linear and multilinear algebra; matrix theory MSC 2000 > 65 Numerical analysis 
MIMS number:  2012.72 
Deposited By:  Nick Higham 
Deposited On:  25 July 2012 
Available Versions of this Item
 Computing the Frechet Derivative of the Matrix Logarithm
and Estimating the Condition Number (deposited 06 August 2013)
 Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number (deposited 16 May 2013)
 Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number (deposited 13 December 2012)
 Computing the Frechet Derivative of the Matrix Logarithm and Estimating the Condition Number (deposited 25 July 2012) [Currently Displayed]
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